SciPost Phys. 2, 005 (2017) ·
published 21 February 2017

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We show that exact time dependent single particle Green function in the
ImambekovGlazman theory of nonlinear Luttinger liquids can be written, for any
value of the Luttinger parameter, in terms of a particular solution of the
Painlev\'e IV equation. Our expression for the Green function has a form
analogous to the celebrated TracyWidom result connecting the Airy kernel with
Painlev\'e II. The asymptotic power law of the exact solution as a function of
a single scaling variable $x/\sqrt{t}$ agrees with the mobile impurity results.
The full shape of the Green function in the thermodynamic limit is recovered
with arbitrary precision via a simple numerical integration of a nonlinear ODE.
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Dr Price: "I thank the referee for their ..."
in Report on Long time thermal asymptotics of nonlinear Luttinger liquid from inverse scattering